# Class 11 Maths NCERT Solutions for Chapter 13 Limits and Derivatives Exercise 13.1

### Limits and Derivatives Exercise 13.1 Solutions

1. Evaluate the Given limit  (x + 3)

Solution

(x + 3) = 3 + 3 = 6

2. Evaluate the Given limit :  (x - 22/7)

Solution

3. Evaluate the Given limit :   Ï€r2.
Solution
Ï€r2 = Ï€(1)2 = Ï€

4. Evaluate the Given limit :  (4x + 3)/(x - 2)
Solution

5. Evaluate the Given limit :  (x10 + x5 + 1)/(x – 1)
Solution

6. Evaluate the Given limit :  [(x + 1)5 - 1]/x
Solution

7. Evaluate the Given limit :   (3x2 - x - 10)/(x2 - 4)
Solution
At x = 2, the value of the given rational function takes the form 0/0 .

8. Evaluate the Given limit  (x4 -81)/(2x2 - 5x - 3)
Solution
At x = 2, the value of the given rational function takes the form 0/0 .

9. Evaluate the Given limit :  (ax + b)/(cx + 1)
Solution

10. Evaluate the Given limit :
Solution

At z = 1, the value of the given function takes the form 0/0 .
Put z1/6 = x so that z →1 as x →1.

11. Evaluate the Given limit (ax2 + bx + c)/(cx2 + bx + a) , a + b + c ≠ 0 .
Solution

= 1 [a + b + c ≠ 0 ]

12. Evaluate the Given limit :
Solution

At x = -2, the value of the given function takes the form  0/0 .

13. Evaluate the Given limit  sin ax/bx
Solution
sin ax/bx
At x = 0, the value of the given function takes the form 0/0 .

14. Evaluate the Given limit : sin ax/sin bx, a,b ≠ 0

Solution

(sin ax/sin bx), a,b ≠ 0
At x = 0, the value of the given function takes the form  0/0.

15. Evaluate the Given limit  [sin(Ï€ -x)]/Ï€(Ï€ - x)
Solution

16. Evaluate the given limit  cos x/(Ï€ - x)
Solution

17. Evaluate the Given limit :   (cos 2x - 1)/(cos x- 1)
Solution
(cos 2x - 1)/(cos x- 1)
At x = 0, the value of the given function takes the form  0/0 .
Now,

18. Evaluate the Given limit  (ax + x cos x)/b sin x
Solution
(ax + x cos x)/b sin x
At x = 0, the value of the given function takes the form 0/0.
Now,

19. Evaluate the Given limit  sec x
Solution

20. Evaluate the Given limit  (sin ax + bx)/(ax + sin  bx) a,b, a+b ≠ 0.
Solution
At x = 0, the value of the given function takes the form 0/0 .
Now,

21. Evaluate the Given limit  (cosec x - cot x)
Solution
At x = 0, the value of the given function takes the form ∞→∞.
Now,

22. Evaluate the Given limit
Solution

At x = Ï€/2 , the value of the given function takes the form  0/0.
Now, put x - Ï€/2 = y so that x → Ï€/2, y → 0.

23. Find f(x)  and  f(x) where
Solution
The given function is

24. Find  f(x) where
Solution
The given function is

25. Evaluate  f(x) where
Solution

It is observed that  f(x) ≠  f(x).
Hence,  f(x) does not exist.

26. Find  f(x) where
Solution
The given function is

It is observed that  f(x) ≠  f(x).
Hence,  f(x) does not exist .

27. Find  f(x) , where f(x) = |x| - 5
Solution
The given functions is f(x) = |x| - 5.

28. Suppose  and if  f(x) = f(1) what are possible values of a and b?
Solution
The given function is

f(x) =  (a + bx) = a + b
f(x) = (b - ax) = b - a
f(1) = 4
It is given that  f(x) = f(1).
∴  f(x) =  f(x) =  f(x) = f(1)
⇒ a + b = 4 and b - a = 4
On solving these two equations, we obtain a = 0 and b = 4.
Thus, the respective possible values of a and b are 0 and 4.

29. Let a1, a2, . . ., an be fixed real numbers and define a function f ( x) = (x − a1) (x − a2)...(x − an). What is f(x) ? For some a ≠ a1, a2, ..., an, compute
Solution
The given function is  f(x) = (x – a1)(x – a2) … (x - an).

30. If   For what value (s)  of a does  f(x) exists ?
Solution
The given function is

When a = 0,

31. If the function f(x) satisfies  [f(x) - 2]/[x2 -1] = Ï€ , evaluate  f(x).
Solution

32. If
For what integers m and n does
f(x) and  f(x) exist ?
Solution
The given function is